Principles of Mathematics in Operations Research

5

Positive Definiteness

Positive definite matrices are of both theoretical and computational impor-

tance in a wide variety of applications. They are used, for example, in opti-

mization algorithms and in the construction of various linear regression mod-

els. As an initiation of our discussion in this chapter, we investigate first the

properties for maxima, minima and saddle points when we have scalar func-

tions with two variables. After introducing the quadratic forms, various tests

for positive (semi) definiteness are presented.

5.1 Minima, Maxima, Saddle points

5.1.1 Scalar Functions

Let us remember the properties for maxima, minima and saddle points when
we have scalar functions with two variables with the help the following exam-
ples.

Fig. 5.1. Plot of f(x, y) = x^2 + y^2